Low Temperature Flowing Gas Plasma

• Low temperature flowing gas plasma is a partially ionized, non-equilibrium state where electrons are much hotter than ions and neutrals, leading to quantized thermodynamic signatures.
• It is characterized by discrete ionization stairs in its equation of state that reflect sequential ionization or dissociation events based on ground-state binding energies.
• The theoretical framework establishes a rigorous basis for quasi-chemical plasma modeling, enabling precise mapping of phase transitions and finite temperature effects in simulations.

A low temperature flowing gas plasma is a partially ionized, non-equilibrium gaseous state of matter maintained by external energy input so that electrons are much hotter than the heavy species (ions, neutrals), with significant effects of gas flow on the transport and reactivity of plasma-generated species. In the limit of extremely low temperature and density, the thermodynamic characterization of such plasmas reveals uniquely quantized features: the formation of “ionization stairs” and associated “thermodynamic spectra,” which govern phase and chemical equilibrium, and establish a rigorous basis for quasi-chemical modeling of plasma thermodynamics (Iosilevskiy, 2010).

1. Limiting Thermodynamic Structure at T → 0, n → 0

When the temperature $T$ and number density $n$ approach zero at a fixed value of the electron chemical potential $(\mu_e)$, the thermodynamic functions of a gaseous plasma exhibit a stepwise (staircase-like) structure in the relevant thermodynamic coordinates:

• The pressure-volume-temperature equation of state (EOS) is given, in reduced units, by

$\frac{PV}{RT} = f(\mu_e)$

where $PV/RT$ traces out plateaus and discontinuous jumps (steps) as $\mu_e$ sweeps over a range.

• The internal energy, corrected for ideal contributions, follows

$U - \frac{3}{2} PV = g(\mu_e)$

with similarly abrupt jumps.

These steps correspond not to thermal excitation but to discrete, ground-state binding energies of atomic, molecular, ionic, and cluster species. The “chemical potential axis” is bounded below by the negative of the major atomic ionization energy and above by a value reflecting the substance’s sublimation energy.

2. “Ionization Stairs” in the Equation of State

The stepped form or “ionization stairs” denotes the precise staircase pattern emergent in both the thermal (compressibility factor $PV/RT$ vs. $\mu_e$) and caloric (corrected internal energy vs. $\mu_e$) EOS in the zero-$T$, zero-$n$ limit. Specifically:

• Plateaus occur between successive critical values of $\mu_e$, each representing the full conversion of one bound complex (atom, molecule, ion) into a more highly ionized (or dissociated) state.
• Steps are located at

$\mu_e = -I_n$

where $I_n$ is the $n$th ionization or dissociation energy (or, for the lower boundary, the cohesive energy such as the sublimation heat).

• For elemental plasmas (e.g., lithium, helium), each step signals the sequential ionization/dissociation event; in mixtures (e.g., solar H–He plasma), the stair structure is superimposed from each constituent, reflecting their unique intrinsic energies.

This structure is generic for any system in the defined limit and is determined exclusively by the set of ground-state binding energies for all possible complexes.

3. Binding Energies and the Intrinsic Energy Scale

Only ground-state binding energies—atomic, ionic, molecular, or cluster energies, along with macroscopic binding energies (e.g., sublimation, dissociation)—appear as “steps” in the limit $T \rightarrow 0$. Excited electronic or vibrational levels do not manifest, ensuring the stepped structure is both sharp and temperature-independent within this asymptotic regime.

This finite, discrete set of energies defines the “intrinsic energy scale” of the system. Every step in the EOS or singular feature in the spectrum of differential parameters (see below) is pinned to one element of this scale.

4. Thermodynamic Differential Properties and “Thermodynamic Spectrum”

Certain thermodynamic response functions—such as the heat capacity ($C_p$) and isentropic coefficient ($Y_s = (\partial \ln P / \partial \ln \rho)_s$)—do not vanish in the zero-temperature limit. Instead, their profiles display sharp, “8-shaped” structures (sometimes referred to as betta-like or “thermodynamic spectra”) as functions of $\mu_e$:

• The heat capacity exhibits sharp emission-like peaks at each binding energy threshold.
• The isentropic coefficient displays corresponding absorption-like dips.

Mathematically, the “lines” of these spectra are perfectly centered at the binding energies in the intrinsic scale. The result is a thermodynamic “spectrum,” analogous in sharpness to spectroscopic line spectra, whose features are entirely determined by ground-state binding energies and not at all by the details of finite-temperature occupation statistics.

5. Mixtures and Universality of the Stepped Structure

The stair-step structure of the limiting EOS and the “thermodynamic spectrum” is universal for any plasma composition, whether elemental or multicomponent:

• In a hydrogen–helium mixture, distinct steps and spectrum features arise at the respective dissociation and ionization energies of each component.
• For plasma of compounds or more complex mixtures, the superimposed stepped structure reflects all ground-state binding energies from all constituents, including those from clusters or solid phases (e.g., sublimation energies).

This universality provides a powerful diagnostic of the intrinsic energy scales present in the plasma and underlines the physical basis for natural phase boundaries and transitions in the low-temperature limit.

6. Zero-Temperature Basis for the Quasi-Chemical (“Chemical Picture”) Approach

The $T \rightarrow 0$ stepped structure establishes a rigorous reference system for deriving the quasi-chemical model of plasma thermodynamics. The asymptotic expansion is performed in temperature at fixed chemical potential:

$Y(P, T) = Y_0(P) + Y_1(P, T) + Y_2(P, T) + \dots$

where:

• $Y_0(P)$ is the limiting, step-structured function corresponding to the “ionization stairs.”
• Corrections $Y_i(P, T)$ are exponentially suppressed as functions of $T$, proportional to $\exp\{-\text{const}(P)/T\}$, and encode finite-temperature population of excited or ionized states.

This expansion—not in density (as in classical chemical approaches) but in temperature—provides a controlled and physically justified way to connect the idealized, ground-state-dominated limit to realistic plasma behavior at finite $T$, thereby reconciling the “physical picture” (based on ions, electrons, and nuclei with Coulomb interactions) with the phenomenological “chemical picture” (involving assemblies of weakly interacting atoms, ions, and molecules).

7. Implications and Applications

The framework elucidated for low temperature flowing gas plasma has multiple implications:

• Phase transitions such as ionization, dissociation, and sublimation in low-temperature plasmas can be precisely mapped to discrete steps in the limiting EOS, avoiding the non-physical, unstable states that can appear in naive extrapolations.
• Finite temperature shell oscillations in plasma equations of state are directly linked as “smoothed counterparts” of the zero-temperature steps, with thermally induced broadening determined by the higher-order terms in the temperature expansion.
• Modelling and simulation efforts for low-temperature plasmas—whether in laboratory discharges, astrophysical environments, or controlled fusion systems—are grounded by this rigorous zero-order reference, yielding analytic control over species populations and elementary reaction channels at the lowest temperatures and densities.

This comprehensive thermodynamic treatment forms the theoretical foundation upon which physically accurate quasi-chemical plasma models and surface/phase transformation predictions are built, particularly relevant for modern developments in plasma physics, materials processing, and planetary/stellar atmosphere studies (Iosilevskiy, 2010).